Two industrial pumps, Pump A and Pump B, are used to fill a storage tank. Working alone at… — GRE General Test

Two industrial pumps, Pump A and Pump B, are used to fill a storage tank. Working alone at its constant standard operating rate, Pump A can fill the empty tank in $12$ hours. Working alone at its constant standard operating rate, Pump B can fill the empty tank in $18$ hours.

To fill the tank, both pumps begin operating simultaneously at their standard rates. After $4$ hours of joint operation, Pump A undergoes maintenance that reduces its operating rate by $25\%$ , while Pump B continues operating at its standard rate. Exactly $2$ hours after Pump A's rate is reduced, Pump B's operating rate is increased by $50\%$ above its standard rate due to a valve adjustment. If both pumps continue operating at these adjusted rates until the tank is full, how many total hours from the initial start does it take to completely fill the tank?

A
$6\frac{2}{3}$ hours
B
$7\frac{1}{7}$ hours
$7\frac{3}{7}$ hoursCevap
D
$8\frac{1}{7}$ hours
E
$8\frac{2}{3}$ hours

Cevap

The total time required to completely fill the tank from the initial start is

7\frac{3}{7}

hours.

The solution proceeds in three stages. In the first 4 hours, both pumps at standard rates completed 5/9 of the job. In the next 2 hours, Pump A operated at 1/16 tank/hr and Pump B at 1/18 tank/hr, completing an additional 17/72 of the job, bringing total completed work to 19/24 of the tank. For the final 5/24 of the tank, Pump A (1/16 tank/hr) and Pump B (1/12 tank/hr) worked at a combined rate of 7/48 tank/hr, taking 10/7 (or 1 3/7) hours. The total time is 4 + 2 + 1 3/7 = 7 3/7 hours.

Adım Adım Çözüm

Determine individual standard rates and combined initial rate.

Pump A rate

r_A = \frac{1}{12}

tank/hr; Pump B rate

r_B = \frac{1}{18}

tank/hr. Initial combined rate

r_1 = \frac{1}{12} + \frac{1}{18} = \frac{5}{36}

tank/hr.

Work rate is the reciprocal of the total time taken to complete one full job.

Calculate work completed in Phase 1 (first 4 hours).

Work completed

= 4 \times \frac{5}{36} = \frac{20}{36} = \frac{5}{9}

of the tank. Remaining work

= 1 - \frac{5}{9} = \frac{4}{9}

of the tank.

Both pumps work at standard combined rate for 4 hours.

Adjust rates for Phase 2 (hours 4 to 6) and calculate work done.

Pump A reduced rate

= 0.75 \times \frac{1}{12} = \frac{1}{16}

tank/hr. Combined rate

r_2 = \frac{1}{16} + \frac{1}{18} = \frac{17}{144}

tank/hr. Work done in 2 hours

= 2 \times \frac{17}{144} = \frac{17}{72}

. Cumulative work

= \frac{5}{9} + \frac{17}{72} = \frac{57}{72} = \frac{19}{24}

of the tank.

Phase 2 lasts 2 hours with Pump A operating at

75\%

efficiency and Pump B at standard rate.

Adjust rates for Phase 3 (from hour 6 onwards) and calculate remaining time.

Pump B increased rate

= 1.50 \times \frac{1}{18} = \frac{1}{12}

tank/hr. Combined rate

r_3 = \frac{1}{16} + \frac{1}{12} = \frac{7}{48}

tank/hr. Remaining work

= 1 - \frac{19}{24} = \frac{5}{24}

. Additional time required

= \frac{5/24}{7/48} = \frac{5}{24} \times \frac{48}{7} = \frac{10}{7} = 1\frac{3}{7}

hours.

Divide remaining fractional work by the new combined rate.

Calculate total elapsed time.

Total time

= 4 + 2 + 1\frac{3}{7} = 7\frac{3}{7}

hours.

Sum the durations of all three phases.

Anahtar Kavram

Multi-stage work-rate problems with variable individual rates and fractional job completion tracking.

Alternatif Yöntem

Define the tank capacity as 144 units (the LCM of 12, 18, 16, 48). Pump A standard rate = 12 units/hr; Pump B standard rate = 8 units/hr. Phase 1 (4 hrs): combined rate = 20 units/hr, work done = 80 units. Phase 2 (2 hrs): Pump A rate = 9 units/hr, Pump B rate = 8 units/hr, combined rate = 17 units/hr, work done = 34 units. Total work done in 6 hrs = 114 units. Remaining work = 30 units. Phase 3: Pump A rate = 9 units/hr, Pump B rate = 12 units/hr, combined rate = 21 units/hr. Additional time = 30/21 = 10/7 hrs. Total time = 6 + 10/7 = 7 3/7 hrs.

Tahmini Süre:2m 30s

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